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  <div class="section" id="numpy-random-binomial">
<h1>numpy.random.binomial<a class="headerlink" href="#numpy-random-binomial" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.random.binomial">
<code class="sig-prename descclassname">numpy.random.</code><code class="sig-name descname">binomial</code><span class="sig-paren">(</span><em class="sig-param">n</em>, <em class="sig-param">p</em>, <em class="sig-param">size=None</em><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.binomial" title="Permalink to this definition">¶</a></dt>
<dd><p>Draw samples from a binomial distribution.</p>
<p>Samples are drawn from a binomial distribution with specified
parameters, n trials and p probability of success where
n an integer &gt;= 0 and p is in the interval [0,1]. (n may be
input as a float, but it is truncated to an integer in use)</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>New code should use the <code class="docutils literal notranslate"><span class="pre">binomial</span></code> method of a <code class="docutils literal notranslate"><span class="pre">default_rng()</span></code>
instance instead; see <em class="xref py py-obj">random-quick-start</em>.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>n</strong><span class="classifier">int or array_like of ints</span></dt><dd><p>Parameter of the distribution, &gt;= 0. Floats are also accepted,
but they will be truncated to integers.</p>
</dd>
<dt><strong>p</strong><span class="classifier">float or array_like of floats</span></dt><dd><p>Parameter of the distribution, &gt;= 0 and &lt;=1.</p>
</dd>
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Output shape.  If the given shape is, e.g., <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">n,</span> <span class="pre">k)</span></code>, then
<code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">k</span></code> samples are drawn.  If size is <code class="docutils literal notranslate"><span class="pre">None</span></code> (default),
a single value is returned if <code class="docutils literal notranslate"><span class="pre">n</span></code> and <code class="docutils literal notranslate"><span class="pre">p</span></code> are both scalars.
Otherwise, <code class="docutils literal notranslate"><span class="pre">np.broadcast(n,</span> <span class="pre">p).size</span></code> samples are drawn.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray or scalar</span></dt><dd><p>Drawn samples from the parameterized binomial distribution, where
each sample is equal to the number of successes over the n trials.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.binom.html#scipy.stats.binom" title="(in SciPy v1.4.1)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.stats.binom</span></code></a></dt><dd><p>probability density function, distribution or cumulative density function, etc.</p>
</dd>
<dt><a class="reference internal" href="numpy.random.Generator.binomial.html#numpy.random.Generator.binomial" title="numpy.random.Generator.binomial"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Generator.binomial</span></code></a></dt><dd><p>which should be used for new code.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The probability density for the binomial distribution is</p>
<div class="math">
<p><img src="../../../_images/math/a3235d9c59e404c34cbb15d54a794631511dfbcc.svg" alt="P(N) = \binom{n}{N}p^N(1-p)^{n-N},"/></p>
</div><p>where <img class="math" src="../../../_images/math/5a939c5280da7202ca4531f175a7780ad5e1f80a.svg" alt="n"/> is the number of trials, <img class="math" src="../../../_images/math/141bbefb74014fc5e43499901bf78607ae335583.svg" alt="p"/> is the probability
of success, and <img class="math" src="../../../_images/math/3bfb3a64189a14b2704f4610827762d5e3145114.svg" alt="N"/> is the number of successes.</p>
<p>When estimating the standard error of a proportion in a population by
using a random sample, the normal distribution works well unless the
product p*n &lt;=5, where p = population proportion estimate, and n =
number of samples, in which case the binomial distribution is used
instead. For example, a sample of 15 people shows 4 who are left
handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4,
so the binomial distribution should be used in this case.</p>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r9ac85fd08ec1-1"><span class="brackets">1</span></dt>
<dd><p>Dalgaard, Peter, “Introductory Statistics with R”,
Springer-Verlag, 2002.</p>
</dd>
<dt class="label" id="r9ac85fd08ec1-2"><span class="brackets">2</span></dt>
<dd><p>Glantz, Stanton A. “Primer of Biostatistics.”, McGraw-Hill,
Fifth Edition, 2002.</p>
</dd>
<dt class="label" id="r9ac85fd08ec1-3"><span class="brackets">3</span></dt>
<dd><p>Lentner, Marvin, “Elementary Applied Statistics”, Bogden
and Quigley, 1972.</p>
</dd>
<dt class="label" id="r9ac85fd08ec1-4"><span class="brackets">4</span></dt>
<dd><p>Weisstein, Eric W. “Binomial Distribution.” From MathWorld–A
Wolfram Web Resource.
<a class="reference external" href="http://mathworld.wolfram.com/BinomialDistribution.html">http://mathworld.wolfram.com/BinomialDistribution.html</a></p>
</dd>
<dt class="label" id="r9ac85fd08ec1-5"><span class="brackets">5</span></dt>
<dd><p>Wikipedia, “Binomial distribution”,
<a class="reference external" href="https://en.wikipedia.org/wiki/Binomial_distribution">https://en.wikipedia.org/wiki/Binomial_distribution</a></p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p>Draw samples from the distribution:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="o">.</span><span class="mi">5</span>  <span class="c1"># number of trials, probability of each trial</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
<span class="go"># result of flipping a coin 10 times, tested 1000 times.</span>
</pre></div>
</div>
<p>A real world example. A company drills 9 wild-cat oil exploration
wells, each with an estimated probability of success of 0.1. All nine
wells fail. What is the probability of that happening?</p>
<p>Let’s do 20,000 trials of the model, and count the number that
generate zero positive results.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">20000</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span><span class="o">/</span><span class="mf">20000.</span>
<span class="go"># answer = 0.38885, or 38%.</span>
</pre></div>
</div>
</dd></dl>

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